X-rays, energy quantization and the Planck’s constant

There are products that use X-rays to make our life better and funnier, such as X-rays glasses to see underneath other people clothes and also many superheroes use them to defeat super villains as doctors and radiologists use to diagnose diseases but what are X-rays? In this post, I want to clarify what they actually are and how, thanks to them, it is possible to demonstrate that the energy is quantized and how the Planck’s constant is precisely calculated. For the latter I will use a few formulas but don’t be afraid, they are easy and not so many.

First of all, let’s do a bit of history.

In 1895, Wilhelm Konrad Röntgen worked in Würzburg, Germany, in a new research field, the cathode rays. Using a cathode rays tube, concretely a Hittorf-Crook tube, covered by a black paper, he observed that a on an indicator paper made of Barium platinocyanide, located beneath the tube, appeared a transversal line when there was a current circulating by the tube. He found this line on the indicator paper strange. On the one hand, according to the research status of the time, the effect could only be due to the radiation of light, but on the other hand, it was impossible that the light was from the tube because of the black paper cover, which didn’t allow the light to go through. Röntgen gave this radiation the name of X-rays because he didn’t know its origin. Almost two months later he had already prepared a communication announcing these results and he even attached a series of pictures that have become famous such as the one of the hand of his wife Anna Bertha Röntgen.

Radiography of the hand of Anna Bertha Röntgen

Röntgen couldn’t give an explanation to the X-rays, but now we know what they are. The cathode rays tube has two electrodes in its opposite sides. One of them, the cathode is heated until it emits electrons and through an electrical potential of some set of tens of thousands volts, the electrons are accelerated towards the electrode on the other side of the tube, the anode. When the electrons hit the anode, it is observed a continuum spectrum of electromagnetic radiation that has wavelengths of 1×10^-10 m. What it actually happens in the anode is that the electron passes close to the nucleus of the atoms of the anode material and they are diverted by the electrical field of the nucleus and they are slowed down and thus they emit radiation (a photon), because when a charged particle it is accelerated or slowed down its speed, in other words its speed changes with time, it emits radiation.

Not all the electrons slow down in the same way, that is, not all of them have the same deceleration because not all of them pass at the same distance of the nucleus. Since they don’t fell the same intensity of the electric field, they don’t get the same deceleration. Because of this, it appears the continuum spectrum. A different value for each electron, and as there are many, the spectrum looks like a continuum.

X-rays continuum spectrum

However an odd phenomenon occurs. For each value of the applied electric potential, it appears a minimum wavelength in the continuum spectrum below which, it is not emitted any radiation. This phenomenon couldn’t be explained with classical physics.

We come now to the second part of the title of this post: energy quantization.

By the end of the XIX century, it was experimentally known that the energy density at a given energy in a frequency interval varied in a way that a low frequencies the bodies emitted such an amount of energy that increased as the frequency increased until it reached a maximum energy and then the energy emitted decreased as the frequency increased. Rayleigh and Jeans tried to explain this energy density distribution but using the physics existing by that time. However they could only explain that the energy increased continuously and that was against the observations. This phenomenon was named ultraviolet catastrophe.

Max Planck proposed that the energy was cuantized, that is there were very small energy packets, named quanta, where each packet had an energy proportional to the frequency. Mathematically this is written as E= hυ, where h is the Planck’s constant. Using this approach, the Rayleigh and Jeans problem was solved and the energy density distribution matched with what was experimentally observed.

Emitted energy as a function of wavelength at different temperatures

Back to the X-rays, the kinetic energy of the electrons is given by their charge e and by the electric potential that accelerates them V, thus:

E=eV

In the case that the electron is totally stopped after interacting with the nucleus and bearing in mind that the energy does neither create nor destroy, that is, the energy before interacting with the nucleus (E=eV) equals the energy of the photon when the electron slows down (E=hυ), we have:

eV=hυ

Resolving for υ and knowing that the speed of light c equals the frequency times the wavelength (c=υλ) we obtain:

λ=hc/eV

We thus obtain a wavelength for each electric potential. Here we can see that something that was not possible to be explained using classical physics, it can be explained using energy quantization, that is, quantum physics.

In the latest formula we see the Planck’s constant.

Now let’s go for the last part of the title of the post: the Planck’s constant.

The speed of light and the charge of the electron have constant and very well known values (c=300000 km/s y e=1,602×10^-19C). When we use a cathode rays tube to generate X-rays, we use a fixed electric potential. If for such electric potential we draw the continuum spectrum of the generated X-rays, we can observe what is the minimum wavelength that enables the generation of X-rays. Therefore, once known the minimum wavelength we can enter all the values in the equation, resolve it for h and establish the value of the Planck’s constant:

h=6,626×10^-34Js

The value of h calculated in this way is very precise due to the precision that we know the rest of the parameters of the equation.

Have the mathematics of this post been painful?

References:

Marie Curie y su tiempo. José Manuel Sanchez Ron

Anna Bertha Roentgen (1832-1919): La mujer detrás del hombre. Daniela García P., Cristián García P. Revista Chilena de Radiología Vol II Nº4, año 2005; 1979-1981

Física Cuántica. Carlos Sanchez del Río (Coordinador)

http://en.wikipedia.org/wiki/Ultraviolet_catastrophe

 

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